There is now a frequent Trenitalia train between Cagliari (Piazza Matteotti station) and the Cagliari Elmas airport, as is shown on these videos:
The train supplements the ARST airport bus, which continues to run every 30 minutes between the airport and the bus depot in Piazza Matteotti:
Seats on the train and bus are all unreserved.
The one-way fare between the airport and central Cagliari is 1.25 Euros on the train and 4 Euros on the bus.
Because virtually all northbound trains will stop at the airport train station, it will now be possible to travel by train to towns north of Cagliari (e.g. Oristano, Sassari and Olbia) without needing to backtrack to downtown Cagliari.